We study constant-cost randomized communication problems and relate them to implicit graph representations in structural graph theory. Specifically, constant-cost communication problems correspond to hereditary graph families that admit constant-size adjacency sketches, or equivalently constant-size probabilistic universal graphs (PUGs), and these graph families are a subset of families that admit adjacency labeling schemes of size O(log n), which are the subject of the well-studied implicit graph question (IGQ). We initiate the study of the hereditary graph families that admit constant-size PUGs, with the two (equivalent) goals of (1) understanding randomized constant-cost communication problems, and (2) understanding a probabilistic version of the IGQ. For each family $\mathcal F$ studied in this paper (including the monogenic bipartite families, product graphs, interval and permutation graphs, families of bounded twin-width, and others), it holds that the subfamilies $\mathcal H \subseteq \mathcal F$ admit constant-size PUGs (i.e. adjacency sketches) if and only if they are stable (i.e. they forbid a half-graph as a semi-induced subgraph). The correspondence between communication problems and hereditary graph families allows for a new method of constructing adjacency labeling schemes. By this method, we show that the induced subgraphs of any Cartesian products are positive examples to the IGQ. We prove that this probabilistic construction cannot be derandomized by using an Equality oracle, i.e. the Equality oracle cannot simulate the k-Hamming Distance communication protocol. We also obtain constant-size sketches for deciding $\mathsf{dist}(x, y) \le k$ for vertices $x$, $y$ in any stable graph family with bounded twin-width. This generalizes to constant-size sketches for deciding first-order formulas over the same graphs.

翻译：我们研究恒定成本随机通信问题，并将其与结构图论中的隐式图表示相关联。具体而言，恒定成本通信问题对应允许常数大小邻接草图（或等价地，常数尺寸概率通用图）的遗传图族，这些图族是允许大小为 O(log n) 的邻接标记方案（即广受研究的隐式图问题主题）的图族的子集。我们开创性研究了允许常数尺寸概率通用图的遗传图族，其双重等价目标为：（1）理解恒定成本随机通信问题；（2）理解隐式图问题的概率化版本。对于本文研究的每个图族 $\mathcal F$（包括单基因二分图族、乘积图、区间图与置换图、有界双宽图族等），子族 $\mathcal H \subseteq \mathcal F$ 允许常数尺寸概率通用图（即邻接草图）当且仅当它们是稳定的（即禁止半诱导子图形式的半图）。通信问题与遗传图族之间的对应关系为构建邻接标记方案提供了新方法。通过该方法，我们证明任意笛卡尔积的诱导子图是隐式图问题的正例。我们证明这种概率构造无法通过使用相等性预言机进行去随机化，即相等性预言机无法模拟 k-汉明距离通信协议。我们还为有界双宽稳定图族中顶点 $x$、$y$ 的 $\mathsf{dist}(x, y) \le k$ 判定问题获得了常数大小草图，该结果可推广至同一图族中一阶公式判定的常数大小草图。